Table REF  lists three examples of triadic resonances and involved modes (see also supplemental Movie 1, Movie 2, Movie 3 for the radial velocity of interacting modes in the equatorial plane) at different control parameters. We can see that the triadic resonance can excite modes with either lower or higher wavenumbers and frequencies with respect to the primary modes, providing a mechanism for the bi-directional energy transfer. For instance, at \(E=10^{-5}\) , \(=0.01\)  and \(Ra=3.3\times 10^6\) , a low frequency (\(\omega /\Omega =0.0918\) ) prograde mode of \(m=3\)  excites a pair of retrograde modes with higher frequencies (\(\omega /\Omega =0.3596\)  and 0.2678) and wavenumbers (\(m=10\)  and 13).  Note that the primary mode in this case shows a multi-cellular pattern of the radial structure [1]}, whereas the primary mode in figure REF  shows a mono-cellular pattern of the radial structure (see supplemental movies). In this case, we also observe cyclic growth and collapse of the unstable modes (figure REF (a-b)), which is a typical behavior of the triadic resonance [2]}, [3]}. We can see that higher wavenumbers modes are excited on top of the \(m=3\)  primary mode during the growth phase (figure REF (c)). As the energy in the growing modes becomes comparable to the zonal flow (\(m=0\) ), the flow suddenly collapses to chaotic small scales (figure REF (d)), known as resonant collapse of inertial modes, which was first observed by [4]} in a precessing cylinder. The collapse is followed by a relaminarization (figure REF ( e), and then the cycle repeats. The whole process is reminiscent of the triadic resonant instability observed in several mechanically driven rotating flows [5]}.
